Quantum Mechanics_ quantum computer

The Bloch sphere is a representation of a qubit, the fundamental building block of

quantum computers.

A quantum computer (also known as a quantum supercomputer) is

acomputation device that makes direct use of quantum-mechanical phenomena, such

as superposition and entanglement, to perform operations on data.[1]Quantum

computers are different from digital computers based on transistors. Whereas digital

computers require data to be encoded into binary digits (bits), each of which is always

in one of two definite states (0 or 1), quantum computation uses qubits (quantum bits),

which can be in superpositions of states. A theoretical model is the quantum Turing

machine, also known as the universal quantum computer. Quantum computers share

theoretical similarities with non-deterministic and probabilistic computers; one

example is the ability to be in more than one state simultaneously. The field of

quantum computing was first introduced by Yuri Manin in 1980[2] and Richard

Feynman in 1982.[3][4] A quantum computer with spins as quantum bits was also

formulated for use as a quantum space–time in 1969.[5]

As of 2014 quantum computing is still in its infancy but experiments have been carried

out in which quantum computational operations were executed on a very small number

of qubits.[6] Both practical and theoretical research continues, and many national

governments and military funding agencies support quantum computing research to

develop quantum computers for both civilian and national security purposes, such

as cryptanalysis.[7]

Large-scale quantum computers will be able to solve certain problems much more

quickly than any classical computer using the best currently known algorithms,

like integer factorization using Shor's algorithm or the simulation of quantum many-

body systems. There exist quantum algorithms, such as Simon's algorithm, which run

faster than any possible probabilistic classical algorithm.[8] Given sufficient

computational resources, however, a classical computer could be made to simulate any

quantum algorithm; quantum computation does not violate theChurch–Turing

thesis.[9]

Basis

A classical computer has a memory made up of bits, where each bit represents either a

one or a zero. A quantum computer maintains a sequence of qubits. A single qubit can

represent a one, a zero, or any quantum superposition of these two qubit states;

moreover, a pair of qubits can be in any quantum superposition of 4 states, and three

qubits in any superposition of 8. In general, a quantum computer with qubits can be

in an arbitrary superposition of up to different states simultaneously (this compares

to a normal computer that can only be in one of these states at any one time). A

quantum computer operates by setting the qubits in a controlled initial state that

represents the problem at hand and by manipulating those qubits with a fixed

sequence of quantum logic gates. The sequence of gates to be applied is called

a quantum algorithm. The calculation ends with a measurement, collapsing the system

of qubits into one of the pure states, where each qubit is purely zero or one. The

outcome can therefore be at most classical bits of information. Quantum algorithms

are often non-deterministic, in that they provide the correct solution only with a

certain known probability.

An example of an implementation of qubits for a quantum computer could start with

the use of particles with two spin states: "down" and "up" (typically written and ,

or and ). But in fact any system possessing an observablequantity A, which

is conserved under time evolution such that A has at least two discrete and sufficiently

spaced consecutive eigenvalues, is a suitable candidate for implementing a qubit. This

is true because any such system can be mapped onto an effective spin-1/2 system.

Bits vs. qubits

A quantum computer with a given number of qubits is fundamentally different from a

classical computer composed of the same number of classical bits. For example, to

represent the state of an n-qubit system on a classical computer would require the

storage of 2n complex coefficients. Although this fact may seem to indicate that qubits

can hold exponentially more information than their classical counterparts, care must

be taken not to overlook the fact that the qubits are only in a probabilistic

superposition of all of their states. This means that when the final state of the qubits is

measured, they will only be found in one of the possible configurations they were in

before measurement. Moreover, it is incorrect to think of the qubits as only being in

one particular state before measurement since the fact that they were in a

superposition of states before the measurement was made directly affects the possible

outcomes of the computation.

Qubits are made up of controlled particles and the means of control (e.g. devices that

trap particles and switch them from one state to another).[10]

For example: Consider first a classical computer that operates on a three-bitregister.

The state of the computer at any time is a probability distribution over the

different three-bit strings 000, 001, 010, 011, 100, 101, 110, 111. If it is a

deterministic computer, then it is in exactly one of these states with probability 1.

However, if it is a probabilistic computer, then there is a possibility of it being in

any one of a number of different states. We can describe this probabilistic state by

eight nonnegative numbers A,B,C,D,E,F,G,H (where A = probability computer is in

state 000, B = probability computer is in state 001, etc.). There is a restriction that

these probabilities sum to 1.

The state of a three-qubit quantum computer is similarly described by an eight-

dimensional vector (a,b,c,d,e,f,g,h), called a ket. Here, however, the coefficients can

have complex values, and it is the sum of the squares of the

coefficients'magnitudes, , that must equal 1. These square

magnitudes represent the probability amplitudes of given states. However, because a

complex number encodes not just a magnitude but also a direction in the complex

plane, the phase difference between any two coefficients (states) represents a

meaningful parameter. This is a fundamental difference between quantum computing

and probabilistic classical computing.[11]

If you measure the three qubits, you will observe a three-bit string. The probability of

measuring a given string is the squared magnitude of that string's coefficient (i.e., the

probability of measuring 000 = , the probability of measuring 001 = , etc..).

Thus, measuring a quantum state described by complex coefficients (a,b,...,h) gives

the classical probability distribution and we say that the quantum

state "collapses" to a classical state as a result of making the measurement.

Note that an eight-dimensional vector can be specified in many different ways

depending on what basis is chosen for the space. The basis of bit strings (e.g., 000,

001, ..., 111) is known as the computational basis. Other possible bases are unit-

length, orthogonal vectors and the eigenvectors of the Pauli-x operator. Ket notation is

often used to make the choice of basis explicit. For example, the state (a,b,c,d,e,f,g,h)

in the computational basis can be written as:

where, e.g.,

The computational basis for a single qubit (two dimensions) is

and .

Using the eigenvectors of the Pauli-x operator, a single qubit is

and .

While a classical three-bit state and a quantum three-qubit state are both eight-

dimensional vectors, they are manipulated quite differently for classical or quantum

computation. For computing in either case, the system must be initialized, for example

into the all-zeros string, , corresponding to the vector (1,0,0,0,0,0,0,0). In

classical randomized computation, the system evolves according to the application

of stochastic matrices, which preserve that the probabilities add up to one (i.e.,

preserve the L1 norm). In quantum computation, on the other hand, allowed

operations are unitary matrices, which are effectively rotations (they preserve that the

sum of the squares add up to one, the Euclidean or L2 norm). (Exactly what unitaries

can be applied depend on the physics of the quantum device.) Consequently, since

rotations can be undone by rotating backward, quantum computations are reversible.

(Technically, quantum operations can be probabilistic combinations of unitaries, so

quantum computation really does generalize classical computation. See quantum

circuit for a more precise formulation.)

Finally, upon termination of the algorithm, the result needs to be read off. In the case

of a classical computer, we sample from the probability distribution on the three-bit

register to obtain one definite three-bit string, say 000. Quantum mechanically,

we measure the three-qubit state, which is equivalent to collapsing the quantum state

down to a classical distribution (with the coefficients in the classical state being the

squared magnitudes of the coefficients for the quantum state, as described above),

followed by sampling from that distribution. Note that this destroys the original

quantum state. Many algorithms will only give the correct answer with a certain

probability. However, by repeatedly initializing, running and measuring the quantum

computer, the probability of getting the correct answer can be increased.

For more details on the sequences of operations used for various quantum algorithms,

see universal quantum computer, Shor's algorithm, Grover's algorithm, Deutsch-Jozsa

algorithm, amplitude amplification, quantum Fourier transform, quantum

gate, quantum adiabatic algorithm and quantum error correction.

Potential

integer factorization is believed to be computationally infeasible with an ordinary

computer for large integers if they are the product of few prime numbers (e.g.,

products of two 300-digit primes).[12] By comparison, a quantum computer could

efficiently solve this problem using Shor's algorithm to find its factors. This ability

would allow a quantum computer to decrypt many of the cryptographicsystems in use

today, in the sense that there would be a polynomial time (in the number of digits of

the integer) algorithm for solving the problem. In particular, most of the popular public

key ciphers are based on the difficulty of factoring integers or the discrete

logarithm problem, which can both be solved by Shor's algorithm. In particular

the RSA, Diffie-Hellman, and Elliptic curve Diffie-Hellman algorithms could be broken.

These are used to protect secure Web pages, encrypted email, and many other types of

data. Breaking these would have significant ramifications for electronic privacy and

security.

However, other cryptographic algorithms do not appear to be broken by these

algorithms.[13][14] Some public-key algorithms are based on problems other than the

integer factorization and discrete logarithm problems to which Shor's algorithm

applies, like the McEliece cryptosystem based on a problem in coding

theory.[13][15] Lattice-based cryptosystems are also not known to be broken by

quantum computers, and finding a polynomial time algorithm for solving

thedihedral hidden subgroup problem, which would break many lattice based

cryptosystems, is a well-studied open problem.[16] It has been proven that applying

Grover's algorithm to break a symmetric (secret key) algorithm by brute force requires

time equal to roughly 2n/2 invocations of the underlying cryptographic algorithm,

compared with roughly 2n in the classical case,[17]meaning that symmetric key lengths

are effectively halved: AES-256 would have the same security against an attack using

Grover's algorithm that AES-128 has against classical brute-force search (see Key

size). Quantum cryptography could potentially fulfill some of the functions of public

key cryptography.

Besides factorization and discrete logarithms, quantum algorithms offering a more

than polynomial speedup over the best known classical algorithm have been found for

several problems,[18] including the simulation of quantum physical processes from

chemistry and solid state physics, the approximation of Jones polynomials, and

solving Pell's equation. No mathematical proof has been found that shows that an

equally fast classical algorithm cannot be discovered, although this is considered

unlikely. For some problems, quantum computers offer a polynomial speedup. The

most well-known example of this is quantum database search, which can be solved

by Grover's algorithm using quadratically fewer queries to the database than are

required by classical algorithms. In this case the advantage is provable. Several other

examples of provable quantum speedups for query problems have subsequently been

discovered, such as for finding collisions in two-to-one functions and evaluating

NAND trees.

Consider a problem that has these four properties:

1. The only way to solve it is to guess answers repeatedly and check them,

2. The number of possible answers to check is the same as the number of inputs,

3. Every possible answer takes the same amount of time to check, and

4. There are no clues about which answers might be better: generating

possibilities randomly is just as good as checking them in some special order.

An example of this is a password cracker that attempts to guess the password for

an encrypted file (assuming that the password has a maximum possible length).

For problems with all four properties, the time for a quantum computer to solve this

will be proportional to the square root of the number of inputs. It can be used to

attack symmetric ciphers such as Triple DES and AES by attempting to guess the secret

key.[19]

Grover's algorithm can also be used to obtain a quadratic speed-up over a brute-force

search for a class of problems known as NP-complete.

Since chemistry and nanotechnology rely on understanding quantum systems, and

such systems are impossible to simulate in an efficient manner classically, many

believe quantum simulation will be one of the most important applications of quantum

computing.[20]

There are a number of technical challenges in building a large-scale quantum

computer, and thus far quantum computers have yet to solve a problem faster than a

classical computer. David DiVincenzo, of IBM, listed the following requirements for a

practical quantum computer:[21]

scalable physically to increase the number of qubits;

qubits can be initialized to arbitrary values;

quantum gates faster than decoherence time;

universal gate set;

qubits can be read easily.

Quantum decoherence

One of the greatest challenges is controlling or removing quantum decoherence. This

usually means isolating the system from its environment as interactions with the

external world cause the system to decohere. However, other sources of decoherence

also exist. Examples include the quantum gates, and the lattice vibrations and

background nuclear spin of the physical system used to implement the qubits.

Decoherence is irreversible, as it is non-unitary, and is usually something that should

be highly controlled, if not avoided. Decoherence times for candidate systems, in

particular the transverse relaxation time T2 (for NMR andMRI technology, also called

the dephasing time), typically range between nanoseconds and seconds at low

temperature.[11]

These issues are more difficult for optical approaches as the timescales are orders of

magnitude shorter and an often-cited approach to overcoming them is opticalpulse

shaping. Error rates are typically proportional to the ratio of operating time to

decoherence time, hence any operation must be completed much more quickly than

the decoherence time.

If the error rate is small enough, it is thought to be possible to use quantum error

correction, which corrects errors due to decoherence, thereby allowing the total

calculation time to be longer than the decoherence time. An often cited figure for

required error rate in each gate is 10−4. This implies that each gate must be able to

perform its task in one 10,000th of the decoherence time of the system.

Meeting this scalability condition is possible for a wide range of systems. However, the

use of error correction brings with it the cost of a greatly increased number of required

qubits. The number required to factor integers using Shor's algorithm is still

polynomial, and thought to be between L and L2, where L is the number of bits in the

number to be factored; error correction algorithms would inflate this figure by an

additional factor of L. For a 1000-bit number, this implies a need for about 104 qubits

without error correction.[22] With error correction, the figure would rise to about

107 qubits. Note that computation time is about L2or about 107 steps and on 1 MHz,

about 10 seconds.

A very different approach to the stability-decoherence problem is to create

atopological quantum computer with anyons, quasi-particles used as threads and

relying on braid theory to form stable logic gates.[23][24]

Developments

There are a number of quantum computing models, distinguished by the basic

elements in which the computation is decomposed. The four main models of practical

importance are:

Quantum gate array (computation decomposed into sequence of few-

qubitquantum gates)

One-way quantum computer (computation decomposed into sequence of one-

qubit measurements applied to a highly entangled initial state or cluster state)

Adiabatic quantum computer or computer based on Quantum

annealing(computation decomposed into a slow continuous transformation of

an initialHamiltonian into a final Hamiltonian, whose ground states contains the

solution)[25]

topological quantum computer[26] (computation decomposed into the braiding

of anyons in a 2D lattice)

The quantum Turing machine is theoretically important but direct implementation of

this model is not pursued. All four models of computation have been shown to be

equivalent to each other in the sense that each can simulate the other with no more

than polynomial overhead.

For physically implementing a quantum computer, many different candidates are being

pursued, among them (distinguished by the physical system used to realize the

qubits):

Superconductor-based quantum computers (including SQUID-based quantum

computers)[27][28] (qubit implemented by the state of small superconducting

circuits (Josephson junctions))

Trapped ion quantum computer (qubit implemented by the internal state of

trapped ions)

Optical lattices (qubit implemented by internal states of neutral atoms trapped

in an optical lattice)

Electrically defined or self-assembled quantum dots (e.g. the Loss-DiVincenzo

quantum computer or[29]) (qubit given by the spin states of an electron trapped

in the quantum dot)

Quantum dot charge based semiconductor quantum computer (qubit is the

position of an electron inside a double quantum dot)[30]

Nuclear magnetic resonance on molecules in solution (liquid-state NMR) (qubit

provided by nuclear spins within the dissolved molecule)

Solid-state NMR Kane quantum computers (qubit realized by the nuclear spin

state of phosphorus donors in silicon)

Electrons-on-helium quantum computers (qubit is the electron spin)

Cavity quantum electrodynamics (CQED) (qubit provided by the internal state of

atoms trapped in and coupled to high-finesse cavities)

Molecular magnet

Fullerene-based ESR quantum computer (qubit based on the electronic spin of

atoms or molecules encased in fullerene structures)

Linear optical quantum computer (qubits realized by processing appropriate

states of different modes of the electromagnetic field through linear optics

elements such as mirrors, beam splitters and phase shifters, e.g.)[31]

Diamond-based quantum computer[32][33][34] (qubit realized by the electronic

or nuclear spin of nitrogen-vacancy centers in diamond)

Bose–Einstein condensate-based quantum computer[35]

Transistor-based quantum computer – string quantum computers with

entrainment of positive holes using an electrostatic trap

Rare-earth-metal-ion-doped inorganic crystal based quantum

computers[36][37](qubit realized by the internal electronic state

of dopants in optical fibers)

The large number of candidates demonstrates that the topic, in spite of rapid progress,

is still in its infancy. But at the same time, there is also a vast amount of flexibility.

In 2001, researchers were able to demonstrate Shor's algorithm to factor the number

15 using a 7-qubit NMR computer.[38]

In 2005, researchers at the University of Michigan built a semiconductor chip that

functioned as an ion trap. Such devices, produced by standard lithographytechniques,

may point the way to scalable quantum computing tools.[39] An improved version was

made in 2006.[citation needed]

In 2009, researchers at Yale University created the first rudimentary solid-state

quantum processor. The two-qubit superconducting chip was able to run elementary

algorithms. Each of the two artificial atoms (or qubits) were made up of a

billion aluminum atoms but they acted like a single one that could occupy two different

energy states.[40][41]

Another team, working at the University of Bristol, also created a silicon-based

quantum computing chip, based on quantum optics. The team was able to runShor's

algorithm on the chip.[42] Further developments were made in 2010.[43]Springer

publishes a journal ("Quantum Information Processing") devoted to the subject.[44]

In April 2011, a team of scientists from Australia and Japan made a breakthrough

in quantum teleportation. They successfully transferred a complex set of quantum data

with full transmission integrity achieved. Also the qubits being destroyed in one place

but instantaneously resurrected in another, without affecting their

superpositions.[45][46]

Photograph of a chip constructed by D-Wave Systems Inc., mounted and wire-bonded

in a sample holder. The D-Wave processor is designed to use

128 superconducting logic elements that exhibit controllable and tunable coupling to

perform operations.

In 2011, D-Wave Systems announced the first commercial quantum annealer on the

market by the name D-Wave One. The company claims this system uses a 128 qubit

processor chipset.[47] On May 25, 2011 D-Wave announced that Lockheed

Martin Corporation entered into an agreement to purchase a D-Wave One

system.[48] Lockheed Martin and the University of Southern California (USC) reached

an agreement to house the D-Wave One Adiabatic Quantum Computer at the newly

formed USC Lockheed Martin Quantum Computing Center, part of USC's Information

Sciences Institute campus in Marina del Rey.[49] D-Wave's engineers use an empirical

approach when designing their quantum chips, focusing on whether the chips are able

to solve particular problems rather than designing based on a thorough understanding

of the quantum principles involved. This approach was liked by investors more than by

some academic critics, who said that D-Wave had not yet sufficiently demonstrated

that they really had a quantum computer. Such criticism softened once D-Wave

published a paper inNature giving details, which critics said proved that the company's

chips did have some of the quantum mechanical properties needed for quantum

computing.[50][51]

During the same year, researchers working at the University of Bristol created an all-

bulk optics system able to run an iterative version of Shor's algorithm. They

successfully factored 21.[52]

In September 2011 researchers also proved that a quantum computer can be made

with a Von Neumann architecture (separation of RAM).[53]

In November 2011 researchers factorized 143 using 4 qubits.[54]

In February 2012 IBM scientists said that they had made several breakthroughs in

quantum computing with superconducting integrated circuits that put them "on the

cusp of building systems that will take computing to a whole new level."[55]

In April 2012 a multinational team of researchers from the University of Southern

California, Delft University of Technology, the Iowa State University of Science and

Technology, and the University of California, Santa Barbara, constructed a two-qubit

quantum computer on a crystal of diamond doped with some manner of impurity, that

can easily be scaled up in size and functionality at room temperature. Two logical qubit

directions of electron spin and nitrogen kernels spin were used. A system which

formed an impulse of microwave radiation of certain duration and the form was

developed for maintenance of protection against decoherence. By means of this

computer Grover's algorithm for four variants of search has generated the right answer

from the first try in 95% of cases.[56]

In September 2012, Australian researchers at the University of New South Wales said

the world's first quantum computer was just 5 to 10 years away, after announcing a

global breakthrough enabling manufacture of its memory building blocks. A research

team led by Australian engineers created the first working "quantum bit" based on a

single atom in silicon, invoking the same technological platform that forms the

building blocks of modern day computers, laptops and phones.[57] [58]

In October 2012, Nobel Prizes were presented to David J. Wineland and Serge

Haroche for their basic work on understanding the quantum world—work which may

eventually help make quantum computing possible.[59][60]

In November 2012, the first quantum teleportation from one macroscopic object to

another was reported.[61][62]

In February 2013, a new technique, boson sampling, was reported by two groups using

photons in an optical lattice that is not a universal quantum computer but which may

be good enough for practical problems. Science Feb 15, 2013

In May 2013, Google Inc announced that it was launching the Quantum Artificial

Intelligence Lab, to be hosted by NASA's Ames Research Center. The lab will house a

512-qubit quantum computer from D-Wave Systems, and the USRA (Universities Space

Research Association) will invite researchers from around the world to share time on it.

The goal is to study how quantum computing might advance machine learning.[63]

In early 2014 it was reported, based on documents provided by former NSA

contractor Edward Snowden, that the U.S. National Security Agency (NSA) is running a

$79.7 million research program (titled "Penetrating Hard Targets") with the aim of

developing a quantum computer capable of breaking encryptionvulnerable to quantum

computers.[64]

Relation to computational complexity theory

The suspected relationship of BQP to other problem spaces.[65]

The class of problems that can be efficiently solved by quantum computers is

called BQP, for "bounded error, quantum, polynomial time". Quantum computers only

run probabilistic algorithms, so BQP on quantum computers is the counterpart

of BPP ("bounded error, probabilistic, polynomial time") on classical computers. It is

defined as the set of problems solvable with a polynomial-time algorithm, whose

probability of error is bounded away from one half.[66] A quantum computer is said to

"solve" a problem if, for every instance, its answer will be right with high probability. If

that solution runs in polynomial time, then that problem is in BQP.

BQP is contained in the complexity class #P (or more precisely in the associated class

of decision problems P#P),[67] which is a subclass of PSPACE.

BQP is suspected to be disjoint from NP-complete and a strict superset of P, but that is

not known. Both integer factorization and discrete log are in BQP. Both of these

problems are NP problems suspected to be outside BPP, and hence outside P. Both are

suspected to not be NP-complete. There is a common misconception that quantum

computers can solve NP-complete problems in polynomial time. That is not known to

be true, and is generally suspected to be false.[67]

The capacity of a quantum computer to accelerate classical algorithms has rigid

limits—upper bounds of quantum computation's complexity. The overwhelming part of

classical calculations cannot be accelerated on a quantum computer.[68] A similar fact

takes place for particular computational tasks, like the search problem, for which

Grover's algorithm is optimal.[69]

Although quantum computers may be faster than classical computers, those described

above can't solve any problems that classical computers can't solve, given enough time

and memory (however, those amounts might be practically infeasible). A Turing

machine can simulate these quantum computers, so such a quantum computer could

never solve an undecidable problem like the halting problem. The existence of

"standard" quantum computers does not disprove theChurch–Turing thesis.[70] It has

been speculated that theories of quantum gravity, such as M-theory or loop quantum

gravity, may allow even faster computers to be built. Currently, defining computation

in such theories is an open problem due to the problem of time, i.e., there currently

exists no obvious way to describe what it means for an observer to submit input to a

computer and later receive output.[71]

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Source: http://wateralkalinemachine.com/quantum-mechanics/?wiki-

maping=Quantum%20computing